Comparing Verboseness for Finite Automata and Turing Machines
A language is called (m,n)-verbose if there exists a Turing machine that enumerates for any n words at most m possibilities for their characteristic string. We compare this notion to (m,n)-fa-verboseness, where instead of a Turing machine a finite automaton is used. Using a new structural diagonalisation method, where finite automata trick Turing machines, we prove that all (m,n)-verbose languages are (h, k)- verbose, iff all (m,n)-fa-verbose languages are (h,k)-fa-verbose. In other words, Turing machines and finite automata behave in exactly the same way with respect to inclusion of verboseness classes. This identical behaviour implies that the Nonspeedup Theorem also holds for finite automata. As an application of the theoretical framework, we prove a lower bound on the number of bits that need to be communicated to finite automata protocol checkers for nonregular protocols.
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